Lemma: Microcausal admission of the tree kernel

$\mathcal{T}$ is a compactly supported distribution on $M\times M$ with

where ${\overline{V}}_{\pm }$ are the closed future/past covector cones including zero. Hence ${\mathsf{T}}_{\mathrm{tree}}\in \mathfrak{W}$: it is a totally Wick-ordered cubic in $\hat{\varphi }$ with a microcausal kernel, in the algebra X-B already used.

Proof. The retarded propagator of a normally hyperbolic operator on Minkowski space satisfies DH

where $(x,\xi )\sim (y,\eta )$ means that $\xi =\eta$ is null, nonzero, and $x-y$ is parallel to ${\xi }^{♯}$. In every element, both covectors are nonzero and $(\xi ,-\eta )$ has one entry in $V_{+}$ and the other in $V_{-}$ whenever either is causal. Derivatives and multiplication by smooth functions do not enlarge the wavefront set, so the local part of $D_1$ composed with ${{\mathsf{E}}_0}^{\mathrm{R}}$ and the kernel $\Lambda$ have the same property. For the nonlocal part, $\mathcal{T}$ is the composition $\Lambda \circ ({{\mathsf{E}}_0}^{\mathrm{R}}g)$ with compact $z$-range by Lemma lem:kernel. Hörmander's composition theorem Hor (Theorem 8.2.14) applies because neither factor has a zero-section component: the exceptional sets ${\mathrm{WF}}_X\times 0$ and $0\times {\mathrm{WF}}_Y$ are empty, and

Along such a chain $\xi =\zeta =\eta$ up to the diagonal parts, so $\xi$ and $\eta$ lie in the same half cone when causal, and $(\xi ,-\eta )$ is never in ${\overline{V}}_{\pm }\times {\overline{V}}_{\pm }$. Compact support is Lemma lem:kernel. The microcausal condition on the kernel of a Wick monomial is X-B's admission criterion BF.